If you choose to paint the bird, try using acrylic fabric paints or even fabric dyes if you have an air brush. They're made to absorb into fabric instead of giving it an unworkable coat that latex paints will. It might help to Google "using fabric paints on fake fur." Cosplayers and Furries do a lot of work coloring their fake fur costumes, and like folks here, they're really happy to show how they do things. I'd watch a few videos and read a few articles before diving right in. And maybe pick up a chunk of fake fur that looks similar to your parrot and try things out before committing it to being painted.
Another thing to think about is just hacking off some of your parrot's "feathers". If you're going to make a zombie bird, roughing up his look by cutting away parts of the perfectly coiffed bird might work. Bobzilla's Davy Jones Parrot is a perfect example of adding a ribcage to the bird to give it a look that says "zombie". While you can't actually cut into the working mechanism, removing portions of the fur and painting the sublayer to look like bone might be very effective. Also as you get him even more worked over, adding a few beat up real feathers where they would be can also make it look as if there are many others missing.